5.5 The Sum and Product Properties

Mkay, so one of the things we learned from Larry's lesson (below) was that sums and products of sinusoids with different periods create new sinusoidal patterns. We learned how to find a graph from a sum or product of sinusoids, and how to do the reverse, to derive these equations from the graph.

One thing we may not have realized that a graph can be both a product of two sinusoids and a sum of two sinusoids.

In 5.5, we find out how to go from a product of a graph to its sum, and vice versa, given one of the equations. (If we're not given an equation, we can figure it out easily given the graph using what we learned from Larry's lesson!) In doing this, we are introduced to the Sum and Product Properties!!!

Here they are:






And, yes, we should memorize them all.

But let's back up a couple of steps-- how did we get these equations in the first place, eh?

example:
Now it's time to get out the good ol' graphing calculator.
In radian mode, por favor, graph the following:
1. y = cos(6x) + cos(4x)
2. y = 2cos(5x) x cos(x) (make this one bold)

Both graphs should be the same. This means that...




Besides the fact that the graph shows this to be true, we can also prove it by expanding the equation using composite argument properties, starting with the left side.







When we add these two equations, we get this:





and when we subtract them, we get this:





WHOA! kinda looks like...






... from earlier! Memorizing the general equations will make life a little bit easier, so that we can say:




instead of going through the long-winded derivation process





Now, let's try to do the reverse, to go from a sum to a product.

example dos:











cool.

I really had a hard time finding a link, but if this helps you (which it probably won't) then I encourage you to use it. http://mathworld.wolfram.com/WernerFormulas.html

on a completely different note, I think for my personalization, I would like to contribute to the 'hip' and 'happening'-ness of this blogger site, If I may. ^_^

This isn't just math-- it's VOGUE math!!! okay, nevermind.

CHRISTINA's TURN!